1. S. Raskhodnikova and A. Smith. Based on slides by C. Leiserson and E. Demaine. 1 Adam Smith L ECTURES 20-21 Priority Queues and Binary Heaps Algorithms and Data Structures CMPSC 465

2. S. Raskhodnikova and A. Smith. Based on slides by C. Leiserson and E. Demaine. 2 Trees Rooted Tree: collection of nodes and edges  Edges go down from root (from parents to children)  No two paths to the same node  Sometimes, children have “ names ” (e.g. left, right, middle, etc)

3. S. Raskhodnikova and A. Smith. Based on slides by C. Leiserson and E. Demaine. Where are trees useful? Some important applications  Parse trees (compilers)  Heaps  Search trees  String search  Game trees (e.g. chess program) E.g. parse tree for 3*(4+2)*(2^6)*( 8/3) 3 3 4 + 2 ^ 2683 / *

4. S. Raskhodnikova and A. Smith. Based on slides by C. Leiserson and E. Demaine. 4 Priority Queue ADT Dynamic set of pairs (key, data), called elements Supports operations:  MakeNewPQ ()  Insert ( S,x ) where S is a PQ and x is a (key,data) pair  Extract-Max ( S ) removes and returns the element with the highest key value (or one of them if several have the same value) Example: managing jobs on a processor, want to execute job in queue with the highest priority Can also get a “ min ” version, that supports Extract-Min instead of Extract-Max Sometimes support additional operations like Delete, Increase-Key, Decrease-Key, etc.

5. S. Raskhodnikova and A. Smith. Based on slides by C. Leiserson and E. Demaine. 5 Heaps: Tree Structure Data Structure that implements Priority Queue  Conceptually: binary tree  In memory: (often) stored in an array Recall: complete binary tree  all leaves at the same level, and  every internal node has exactly 2 children Heaps are nearly complete binary trees  Every level except possibly the bottom one is full  Bottom layer is filled left to right

6. S. Raskhodnikova and A. Smith. Based on slides by C. Leiserson and E. Demaine. 6 Height Heap Height =  length of longest simple path from root to some leaf  e.g. a one-node heap has height 0, a two- or three-node heap has height 1,... Exercises:  What is are max. and min. number of elements in a heap of height h? ans: min = 2 h, max = 2 h +1 -1  What is height as a function of number of nodes n? ans: floor( log(n) )

7. S. Raskhodnikova and A. Smith. Based on slides by C. Leiserson and E. Demaine. 7 Array representation Instead of dynamically allocated tree, heaps often represented in a fixed-size array  smaller constants, works well in hardware. Idea: store elements of tree layer by layer, top to bottom, left to right Navigate tree by calculating positions of neighboring nodes:  Left(i) := 2i  Right(i) := 2i+1  Parent(i) := floor(i/2) Example: [20 15 8 10 7 5 6]

8. S. Raskhodnikova and A. Smith. Based on slides by C. Leiserson and E. Demaine. 8 Review Questions Is the following a valid max-heap?  20 10 4 9 6 3 2 8 7 5 12 1  (If not, repair it to make it valid) Is an array sorted in decreasing order a valid max- heap?

9. S. Raskhodnikova and A. Smith. Based on slides by C. Leiserson and E. Demaine. 9 Local Repair: Heapify “Down” Two important “ local repair operations ” :  Heapify “Down”, Heapify “ Up ” Suppose we start from a valid heap and decrease the key of node i, so  it is still smaller than parent  the two subtrees rooted at i remain valid How do we rearrange nodes to make the heap valid again?  Idea: let new, smaller key “ sink ” to correct position Find index with largest key among { i, Left( i ), Right( i )} If i != largest, EXCHANGE i with largest and recurse on largest

10. S. Raskhodnikova and A. Smith. Based on slides by C. Leiserson and E. Demaine. 10 Local Repair: Heapify “Down” Pseudocode in book Running time: O(height) = O(log n)  Tight in worst case Correctness: by induction on height

11. S. Raskhodnikova and A. Smith. Based on slides by C. Leiserson and E. Demaine. 11 MAX-HEAPIFY(-DOWN) Exercise: what does Max-Heapify do on the following input?  2.5 10 4 9 6 3 2 8 7 5 0 1 This gives us our first PQ operation: Extract-Max(A) 1. tmp := A[1] 2. A[1] := A[heap-size] 3. heap-size := heap-size-1 4. MAX-HEAPIFY-DOWN(A,1)  return tmp

12. S. Raskhodnikova and A. Smith. Based on slides by C. Leiserson and E. Demaine. 12 Local Repair: Heapify “Up” Two important “ local repair operations ”  Heapify “ Down ”  Heapify “Up” Suppose we start from a valid heap and increase the key of node i, so  it is still larger than both children, but  might be larger than parent How do we rearrange nodes to make the heap valid again?  Idea: let new, larger key “ float ” to right position If A[i] > A[Parent(i)], EXCHANGE i with parent and recurse on parent Pseudocode in CLRS Chap 6.2: “ MAX-HEAPIFY ”

13. S. Raskhodnikova and A. Smith. Based on slides by C. Leiserson and E. Demaine. 13 Local Repair: Heapify “Up” Pseudocode in book (see “ Increase-key ” ) Running time: O(log n)  tight in worst case Correctness: by induction on height

14. S. Raskhodnikova and A. Smith. Based on slides by C. Leiserson and E. Demaine. 14 MAX-HEAPIFY(-UP) Exercise: what does Max-Heapify do on the following input?  20 10 4 9 6 3 2 8 7 21 0 1 This gives us our second PQ operation: Insert(A,x) 1. A[heap-size+1] := x 2. heap-size := heap-size+1 3. MAX-HEAPIFY-UP(A,heap-size)

15. S. Raskhodnikova and A. Smith. Based on slides by C. Leiserson and E. Demaine. 15 Other PQ operations Max(S): return max element without extracting it Increase-Key(S,i,new-key) Decrease-Key(S,i,new-key)  Both of these require knowing position of desired element  Can be taken care by “ augmenting ” using a dictionary data structure, such as hash table Delete(S,i)  Delete the element in position i

16. S. Raskhodnikova and A. Smith. Based on slides by C. Leiserson and E. Demaine. 16 Heap Sort Heaps give an easy O(n log n) sorting algorithm:  For i=2 to n Insert(A,A[i])  For i= n downto 3 A[i] := Extract-Max(A) There is a faster way (O(n) time) to build a heap from an unsorted array.

17. S. Raskhodnikova and A. Smith. Based on slides by C. Leiserson and E. Demaine. 17 Search Can we search for an element in a heap in sublinear (that is, o(n)) time, in the worst case? How would you prove that such a search algorithm does not exist?

18. S. Raskhodnikova and A. Smith. Based on slides by C. Leiserson and E. Demaine. Heaps: Review Heap-leap-jeep-creep(A): 1. A.heap-size = 0 2. For i=1 to len(A) tmp = A[i] Heap-Insert(A,tmp) 3. For i=len(A) down to 1 tmp = Heap-Extract-Max(A) A[i]=tmp What does this do? What is the running time of this operation? 18